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REVIEW 4 major objections 4 minor 3 references

The paper derives noise thresholds under which adaptive circuits (mid-circuit measurement plus classical feedforward) beat standard ones for GHZ and W-state preparation, then shows current hardware misses them.

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Under a worst-case noise model, adaptive GHZ and W-state preparation can exponentially beat non-adaptive circuits only in narrow gate-vs-idling error regimes, and current IBM hardware is too noisy and too restricted to realize that advantage.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection GHZ noise-threshold analysis is solid and worth refereeing; the W-state theorem has an exponent error and the hardware experiment tests a degraded circuit, so the headline conclusion overreaches. the 4 major comments →

arxiv 2509.06455 v1 pith:BPUYEKYQ submitted 2025-09-08 quant-ph

Theoretical and experimental analysis of adaptive quantum computers

Niels M. P. Neumann This is my paper
classification quant-ph MSC 81P68 PACS 03.67.Ac
keywords adaptive quantum computingquantum feedforwardmid-circuit measurementworst-case noise modelGHZ state preparationW-state preparationconstant-depth state preparationsuccess probability analysis
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Adaptive quantum algorithms replace some quantum work with intermediate classical computation: mid-circuit measurements feed forward into later gates, and the paper asks when that substitution actually helps under noise. It introduces a worst-case error model in which any error is a random unitary that never cancels, making the success probability of a circuit an exact product of per-operation success probabilities, and it uses that model to compare adaptive and standard circuits for preparing GHZ and W states. The resulting theorems give explicit crossover thresholds: adaptive GHZ preparation wins over the standard all-to-all circuit once the two-qubit success probability p_d exceeds the idle-coherence probability p_id to a power that grows like log n, and over a linear-neighbour circuit once that power grows like n; for W states the required power is Ω(n/(log n log log n)). The paper then runs the GHZ protocols on a 127-qubit superconducting processor, where the formulas predict an adaptive advantage starting at about 15 qubits, and finds instead that the approaches perform similarly for larger n, with noise dominating. The paper's conclusion, stated plainly, is that adaptive quantum algorithms have potential but currently do not outperform full quantum algorithms.

Core claim

On the paper's own terms, the central claim is a pair of threshold comparisons proved under the worst-case error model of Section 2. For an n-qubit GHZ state, adaptive preparation beats the standard all-to-all circuit whenever the two-qubit success probability satisfies p_d ≳ (1+ε) p_id^{n/(n−1)(⌈log2 n⌉/2−2)}, and beats the linear-neighbour standard circuit whenever p_d ≳ (1+ε) p_id^{n/(n−1)(⌈n/2⌉/2−2)}; these simplify to p_d ≳ p_id^{Ω(log n)} and p_d ≳ p_id^{Ω(n)}. For W states, Theorem 2 gives the analogous condition p_d ≳ (1+ε) p_id^{3n/(59 log n log log n)}. The experimental part implements the GHZ protocols on a 127-qubit superconducting processor; although the formulas predict the ada

What carries the argument

The machinery is the worst-case error model plus product-form success probabilities. Every operation—single-qubit gate, two-qubit CNOT, measurement, intermediate classical computation, and every idling period—is assigned a success probability (p_s, p_d, p_m, p_ic, p_id, p_im), and because errors are Haar-random unitaries that never cancel, the total success probability of a protocol is the product of these factors. The argument reduces the comparison of two protocols to an inequality between powers of p_d and p_id, from which the threshold exponents of Theorems 1 and 2 are read off. For the W state, the adaptive protocol is decomposed into fanout gates, an OR-reduction, an exponential-size O

Load-bearing premise

The whole derivation hinges on the assumption that an error scrambles the quantum state completely and that two errors never cancel each other; real noise channels include identity components and partial cancellation, and if that assumption fails the derived thresholds are not guaranteed to transfer.

What would settle it

On a device where two-qubit gate and idle success probabilities are calibrated, choose n so that p_d ≈ (1+ε) p_id^{n/(n−1)(⌈log2 n⌉/2−2)}, prepare GHZ states with the all-to-all and adaptive protocols, and compare measured success probabilities. If the adaptive-to-standard ratio does not reach (1+ε)^{2(n−1)}, the threshold bound is not transferable to that hardware. A more direct check of the model is to apply two consecutive gates to the same qubit and test whether the identity outcome has nonzero probability: under the Haar-random-error assumption the probability is exactly zero, while any d

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the model and thresholds are right, adaptive GHZ preparation is a winning strategy only on hardware whose idle-coherence time is much longer than its gate time—roughly, p_d exceeding p_id to a power that grows like log n for all-to-all connectivity or like n for linear-neighbour connectivity.
  • Once p_d and p_id are calibrated for a device, the formulas predict a crossover system size; on the paper's calibration the crossover sits near n=15, but the measured data do not show the predicted separation, indicating that overheads such as limited classical feedforward and parallel-gate constraints move the practical crossover.
  • For W states, the adaptive constant-depth protocol carries enough subroutine overhead—qubits, fanouts, OR circuits—that its advantage requires p_d to exceed p_id to the power Ω(n/(log n log log n)), a much stricter condition than for GHZ states.
  • The hybrid strategy—prepare small GHZ blocks with standard circuits and join them adaptively—inherits the exponential advantage under weaker conditions than fully adaptive preparation, giving a practical interpolation between the two regimes.
  • On current hardware, neither protocol reliably produces large GHZ states: for n≥30 both approaches return distributions spread across many Hamming weights, so error-suppressed or fault-tolerant state preparation is still needed for scale-up.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The paper's worst-case model is conservative, so the derived thresholds are sufficient conditions rather than necessary ones; on realistic depolarizing or dephasing noise, where identity components make error cancellation possible, adaptive advantage might appear at less extreme p_d/p_id ratios, and the product-form derivation would need to be re-run channel by channel.
  • The same comparison template transfers to other constant-depth state-preparation targets from the shallow-circuit feedforward literature—Dicke states, topological states, and related constructions—once each subroutine has a product-form success probability.
  • The experiment's main bottleneck may not be quantum noise alone but the classical-feedforward instruction set, which forced one correction branch per measurement outcome instead of a prefix-parity computation; devices that support arbitrary mid-circuit classical logic at low latency could realize the theoretical advantage at smaller system sizes.
  • A concrete follow-up, partially anticipated in the paper's discussion, is to re-derive the thresholds with a parallelism factor; on crosstalk-limited chips that restrict simultaneous gates, the dense adaptive circuits stretch in time, effectively raising the p_id exponents and pushing the crossover to larger n.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper analyzes, theoretically and experimentally, whether adaptive quantum algorithms (using intermediate classical computations) can outperform non-adaptive quantum algorithms for state preparation. It introduces a worst-case noise model in which every error is a Haar-random unitary applied instead of the intended gate, so error-cancellation has probability zero and the total success probability factorizes into per-operation success probabilities. On this basis, Section 3 derives success probabilities for GHZ-state preparation via an all-to-all, linear-nearest-neighbor, adaptive, and hybrid approach, and proves Theorem 1, giving thresholds on the CNOT success probability p_d relative to the idling success probability p_id under which the adaptive protocol outperforms the standard protocols. Section 5 performs a similar analysis for W-state preparation, leading to Theorem 2. The paper also implements the GHZ protocols on IBM Brisbane (4096 shots per circuit, n up to 55) and compares the qualitative behavior of measured distributions with theory. The abstract's headline conclusion is that adaptive quantum algorithms currently do not outperform full quantum algorithms.

Significance. If the theoretical results are correct, they provide a concrete, falsifiable condition—expressible in terms of gate-error and idling-error rates—for when constant-depth adaptive state preparation should beat standard circuits. The GHZ derivation (Equations (2)-(13)) is internally consistent, and the paper is commendable for deriving explicit success-probability formulas rather than relying on asymptotic circuit-size arguments alone. The experimental section provides reproducible code and calibration data. However, the experimental support for the headline claim is substantially weakened by the paper's own admission that the implemented adaptive circuit is not the protocol analyzed in the theorems, and by the absence of a quantitative comparison between measured and predicted success probabilities. The W-state theorem also contains an exponent error in its statement. The theoretical framework is a useful first step, but the manuscript needs revision before the empirical conclusion can be accepted.

major comments (4)
  1. [Section 4.1, 4.2, and Abstract] The experimental circuit is not the constant-depth adaptive protocol analyzed in Theorem 1. The code shown in Section 4.1 applies one conditional X per measured ancilla (if_test on crx[i], with X on all qrm[j] for j>i), so up to n-1 sequential correction layers are used. The paper itself states that 'the resulting implementation thus is suboptimal, using significantly more gates than initially required.' Theorem 1, by contrast, is for the Buhrman et al. constant-depth protocol with a single prefix-sum correction. Since Table 2 satisfies Inequality (11) for n≥15, the paper's own theory predicts the faithful adaptive protocol should outperform the standard protocol on this device. That the degraded implementation did not do so tests only the feedforward-restricted variant, not 'adaptive quantum algorithms' generally. The abstract's final sentence therefore overreaches. The experiments shou
  2. [Theorem 2 and Section 5.4] The gain exponent in Theorem 2 is inconsistent with the proof. The theorem states P(W,adaptive) ≳ (1+ε)^{59n/(log2 n log2 log2 n)} P(W,linear), but the derivation in Section 5.4 gives (1+ε)^{59 n log2 n log2 log2 n}: the threshold exponent is 3n/(59 log2 n log2 log2 n), and substituting p_d = (1+ε) times that threshold into the leading term p_d^{59nkt} yields (1+ε)^{59nkt} with kt = log2 n log2 log2 n. The theorem should state the product, not the quotient, in the exponent. This is a central claim, so the error must be fixed.
  3. [Section 4.2] The paper says the hardware results are compared with theoretical results from Equations (3) and (4), but no quantitative comparison is reported. The figures show measurement counts and Hamming-weight histograms, but there is no table or plot of the measured success probability versus the predicted P(GHZ_n,linear) and P(GHZ_n,adaptive) for the different n. Since only 4096 samples are used, confidence intervals are also absent. Without a quantitative comparison, the statement that the implementations 'compare the outcomes to our derived theoretical results' is unsupported. Please add a table with measured success probabilities and the corresponding theoretical predictions, or explicitly state that only qualitative trends are compared.
  4. [Section 2 and Section 6] The worst-case error model (Haar-random errors, zero cancellation probability) is a strong idealization, and the paper acknowledges that it 'most likely overestimates the impact of errors.' This is acceptable for a first-order theoretical model, but the model is also used to compute absolute numerical success probabilities in Section 4.2 (e.g., P(Brisbane, GHZ55,adaptive) ≥ 4.82e-2). Given that the hardware violates several model assumptions—gate decomposition, crosstalk, parallel-gate limits, and partial error cancellation—the numerical predictions should not be presented as quantitative expectations. The paper should state clearly that these numbers are lower bounds under an idealized model and are not directly testable on current hardware. Section 6 already lists these caveats, but they need to be reflected in Section 4's claims.
minor comments (4)
  1. [Throughout] There are several typos and grammatical issues, e.g., 'adapative' in Section 6, 'Measurements results' in the Figure 5 caption. A careful proofreading pass is needed.
  2. [Equation (24)] The notation P(cR_Z) is used but controlled-R_Z is not defined in the text. Please define the gate and its decomposition, or refer to a standard source.
  3. [Section 5.4] The proof of Theorem 2 uses the approximations ⌈x⌉ ≈ x ≈ ⌊x⌋ and 2^t ≈ k without quantifying the error. Since the theorem is stated for all n=2^k, please state the asymptotic regime (e.g., n→∞) and specify that the comparison is up to leading-order terms, as the phrase 'with respect to the most significant terms' already indicates.
  4. [Section 4.1] The code listing contains placeholders like '<your token here>', which is expected, but there is no data-availability statement or repository link for the experimental data. For reproducibility, please provide the raw measurement counts or a public repository.

Circularity Check

0 steps flagged

No significant circularity: the theoretical thresholds are derived algebraically from circuit counts, and the experimental comparison is a genuinely out-of-sample check.

full rationale

The paper's derivation chain is self-contained: it defines per-operation success probabilities (p_s, p_d, p_m, p_id, etc.) under an explicit worst-case noise model, counts the gates and idling intervals in each circuit, and multiplies the corresponding success probabilities. The thresholds in Theorem 1 and Theorem 2 are obtained by writing P(adaptive) >= P(non-adaptive) and solving for p_d as a function of p_id; they are algebraic rearrangements of the derived expressions (Eqs. (2)-(4) and Eqs. (33)-(35)), not conclusions that were assumed from the start. The experimental section uses a separate calibration snapshot (Table 2) to compute expected success probabilities and then compares them with fresh measurements on IBM Brisbane; the fact that the hardware results contradict the model-based expectation for n >= 15 is evidence against fitting, not for circularity. The heavy reliance on [Buh+24], co-authored by the present author, is transparent and the protocol is used as an object of analysis rather than as a black-box uniqueness theorem; the paper re-derives the relevant success probabilities from the circuits. The acknowledged mismatch between the ideal adaptive protocol and the feedforward-restricted implementation is a real external-validity limitation, but it does not make any step circular. No fitted parameter is renamed as a prediction, and no claim reduces by definition to its own inputs. Therefore the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 9 axioms · 0 invented entities

All quantitative inputs (p_s, p_d, p_id, p_m, p_im, p_ic) come from a single IBM calibration snapshot (Table 2) and enter the derivations as external inputs rather than fitted parameters. No quantity is tuned to make the experimental results match the theory; on the contrary, the hardware data contradict the model-based prediction for n≥15. The main intellectual debt is the worst-case error model and the cited adaptive constructions, both stated explicitly.

axioms (9)
  • domain assumption Any error applies a Haar-random unitary, making error cancellation probability zero; total success probability factorizes over operations (Section 2)
    This is the worst-case model that all theoretical comparisons rely on; the paper acknowledges it overestimates real hardware errors.
  • domain assumption Measurement errors flip the reported outcome and cannot be undone; p_m counts only the probability of a correct readout (Section 2)
    Needed for the adaptive circuits, where a wrong measurement leads to wrong feedforward.
  • domain assumption Intermediate classical computations always succeed: p_c=1 (Section 2, Table 1)
    Simplifies the expressions; the hardware section later shows the real limitation is classical expressiveness rather than classical accuracy.
  • domain assumption Relative error magnitudes: p_s≈1≈p_is, p_d≈p_m, p_id≈p_im≈p_ic (Section 2)
    Used to reduce the comparison inequalities to two-variable thresholds (p_d vs p_id); motivated by IBM calibration data.
  • domain assumption Continuous gate set, so no approximation error from discretizing gates (Section 1)
    Required because the worst-case model treats approximation error as a fatal error.
  • domain assumption Unlimited parallel gate application in the theoretical circuits (Section 6)
    The adaptive circuits are dense; parallel-gate limits on real devices lengthen idle times and reduce the advantage, as the discussion admits.
  • domain assumption Correctness of cited subroutines: adaptive GHZ and W-state protocols of [Buh+24], OR-reduction of [HŠ05], OR-gate of [TT13], and distributed GHZ/fanout construction of [YL04]
    All success-probability derivations are built on these constructions without re-deriving them.
  • standard math Haar measure integration: for random unitaries D,B and fixed unitaries G, P(D G B = G)=0 (Section 2)
    Standard property of Haar measure; used to justify the no-cancellation claim.
  • domain assumption Asymptotic approximations in the W-state proof: ⌈x⌉≈x, 2^t≈k, and keeping only leading terms preserves the inequality direction (Section 5.4)
    This is how Theorem 2 is obtained; no error bounds for the dropped terms are given.

reviewed 2026-08-04 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Theoretical and experimental analysis of adaptive quantum computers." pith.science (2026). https://pith.science/paper/BPUYEKYQ

@misc{pith2026250906455,
  author       = {Pith},
  title        = {Pith review of: Theoretical and experimental analysis of adaptive quantum computers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BPUYEKYQ}},
  note         = {Machine review of arXiv:2509.06455}
}
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abstract

Fault-tolerant quantum computations require alternating quantum and classical computations, where the classical computations prove vital in detecting and correcting errors in the quantum computation. Recently, interest in using these classical computations has been growing again, not to correct errors, but to perform computations. Various works have looked into these so-called adaptive quantum algorithms. Few works however have looked in the advantages of adaptive quantum algorithms in realistic scenarios. This work provides the first step in this direction. We introduce a worst-case noise model and use it to derive success probabilities for preparing a GHZ state and preparing a $W$-state using either an adaptive quantum algorithm, or using a standard non-adaptive quantum algorithm. Next, we implemented these protocols on quantum hardware and we compare the outcomes to our derived theoretical results. We find that despite their potential, adaptive quantum algorithms currently do not outperform full quantum algorithms.

Figures

Figures reproduced from arXiv: 2509.06455 by Niels M. P. Neumann.

Figure 1
Figure 1. Figure 1: Circuit for preparing the GHZ state using an all-to-all connectivity [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Circuit for preparing the GHZ state using a linear nearest-neighbor [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: GHZ state preparation using an adaptive quantum circuit. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Measurement results for preparing a GHZ state on few qubits on the [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Measurements results for preparing a GHZ state on [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Measurement results for preparing a GHZ state on [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Implementation of a quantum fanout gate with the GHZ state prepa [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: An adaptive circuit that prepares the state [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Exact circuit for preparing the W-state for n = 4. Every gate parametrized by 1/n denotes a controlled-RY -gate with argument θ = −2 arccos p 1/n. for the OR-gate given in Equation (25). We use t = ⌈log2 (k + 1)⌉ and obtain P(W, adaptive) = P(Uncompressn)P(Compressn) ≥ p 4nk+3k s p 3nk+7n−3k is P 4k F anoutn P n F anoutk+1 P 4n iF anoutP n ORk ≥ p 22nkt+14nk+2n⌈(2t−1)/2⌉+n(3·2 t+⌈k/2⌉)+2nt(5·2 t−2)+3k+4k⌈(… view at source ↗
Figure 10
Figure 10. Figure 10: Exact decomposed circuit for preparing the [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages

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    Pith/arXiv arXiv 2024

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.